On approximating the eigenvalues and eigenvectors of linear continuous operators

We consider the computing of an eigenpair (an eigenvector $v=(v^{(i)})_{i=1,n}$ and an eigenvalue $\lambda$) of a matrix $A\in\mathbb{R}^{n\times n}$, by considering a supplementary condition (we call it norming function for the eigenvector), represented by a polynomial of degree 2. A usual choice is
$$\begin{equation} G(v):={\textstyle\frac12} \sum_{i=1}^n( v^{(i)})^2-1=0. \end{equation}$$
We propose here a new choice:
$$\begin{equation} G(v):={\textstyle\frac1{2n}} \sum_{i=1}^n( v^{(i)})^2-1=0, \end{equation}$$
which has the advantage that leads to a smaller nonlinearity.

Indeed, for the $n+1$-dimensional nonlinear system (a polynomial equation of degree 2) we obtain:
$$F\left( x\right) :={{Av-\lambda v} \choose {G(v)-1}} =0,$$
and our proposed choice leads to a second derivative of $F$ that has norm 2 (compared to $n$, for the usual choice).

We consider this problem in a general setting, of a linear continuous operator $A:V\rightarrow V$, $V$ Banach space, which leads to a polynomial equation of degree two.

We study the semilocal convergence of an iterative method of Schultz type for this problem; this method has the advantage that does not require the solving of a linear system at each iteration step:
$$\begin{align*} x_{k+1} =&x_k-\Gamma_kF\left( x_k\right) \\ \Gamma_{k+1} =&\Gamma_k\left( 2I-F^{\prime }\left( x_{k+1}\right) \Gamma_k\right) , \qquad k=0,1,\ldots, \end{align*}$$
where $x_0\in X$, $\Gamma_0\in \mathcal{L}\left( X\right)$, and $I$ is the identity operator of $\mathcal{L}\left( X\right)$.

We obtain semilocal convergence conditions which show that the method has r-convergence order 2.

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

linear continuous operator; Banach space; Newton type method; eigenvector; eigenvalue; eigenpair; Schultz method; r-convergence order.

E. Cătinaş, I. Păvăloiu, On approximating the eigenvalues and eigenvectors of linear continuous operators, Rev. Anal. Numér. Théor. Approx., 26 (1997) nos. 1-2, pp. 19-27.

Scanned paper: on the journal website.

Latex-pdf version of the paper. (soon)

Google Scholar citations